Sparse representations in a certain transform basis or dictionary have been exploited in compressive sensing to reconstruct high-dimensional signals from few samples. For compressive spectral video sensing, sparse representations based on the Kronecker product of one-dimensional transforms have been traditionally used. However, most of the sparse and recovery models are posed in vector spaces leading to the vectorization of high-dimensional spectral videos, where the intrinsic structure of the signal is not exploited. This work presents an approach based on higher-order sparse signal representation for spectral-video reconstruction. For this, the coded aperture snapshot spectral imager is employed to acquire the video. Then, the inverse problem is modified to sparsely represent the signal as a four-dimensional array, where, on each iteration of the recovery process, the basis is refined using the projected estimation. Simulations over three spectral videos show an improvement of up to 5 dB in terms of PSNR using the proposed approach compared to traditional sparse bases.